Integrand size = 27, antiderivative size = 125 \[ \int \frac {(-1+x) (1+3 x)}{(-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} \left (124616800+1235276981 x+1609127381 x^2\right )+612314840 \sqrt {3} (-1+x) \sqrt [3]{-x+x^3}+2605939922 \sqrt {3} \left (-x+x^3\right )^{2/3}}{-39304000+3108349623 x+2990437623 x^2}\right )-\frac {1}{2} \log \left (\frac {-1+3 x+3 (-1+x) \sqrt [3]{-x+x^3}-3 \left (-x+x^3\right )^{2/3}}{-1+3 x}\right ) \] Output:
-3^(1/2)*arctan((612314840*3^(1/2)*(x^3-x)^(1/3)*(-1+x)+3^(1/2)*(160912738 1*x^2+1235276981*x+124616800)+2605939922*3^(1/2)*(x^3-x)^(2/3))/(299043762 3*x^2+3108349623*x-39304000))-1/2*ln((3*(-1+x)*(x^3-x)^(1/3)+3*x-3*(x^3-x) ^(2/3)-1)/(-1+3*x))
\[ \int \frac {(-1+x) (1+3 x)}{(-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=\int \frac {(-1+x) (1+3 x)}{(-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx \] Input:
Integrate[((-1 + x)*(1 + 3*x))/((-1 + 3*x)*(-x + x^3)^(2/3)),x]
Output:
Integrate[((-1 + x)*(1 + 3*x))/((-1 + 3*x)*(-x + x^3)^(2/3)), x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(x-1) (3 x+1)}{(3 x-1) \left (x^3-x\right )^{2/3}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \left (x^2-1\right )^{2/3} \int \frac {(1-x) (3 x+1)}{(1-3 x) x^{2/3} \left (x^2-1\right )^{2/3}}dx}{\left (x^3-x\right )^{2/3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 x^{2/3} \left (x^2-1\right )^{2/3} \int \frac {(1-x) (3 x+1)}{(1-3 x) \left (x^2-1\right )^{2/3}}d\sqrt [3]{x}}{\left (x^3-x\right )^{2/3}}\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {3 (-x-1)^{2/3} (1-x)^{2/3} x^{2/3} \int \frac {\sqrt [3]{1-x} (3 x+1)}{(1-3 x) (-x-1)^{2/3}}d\sqrt [3]{x}}{\left (x^3-x\right )^{2/3}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 (-x-1)^{2/3} (1-x)^{2/3} x^{2/3} \int \left (\frac {2 \sqrt [3]{1-x}}{(1-3 x) (-x-1)^{2/3}}-\frac {\sqrt [3]{1-x}}{(-x-1)^{2/3}}\right )d\sqrt [3]{x}}{\left (x^3-x\right )^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 (-x-1)^{2/3} (1-x)^{2/3} x^{2/3} \left (\frac {4}{9} \int \frac {1}{\left (\sqrt [3]{-3} \sqrt [3]{x}+1\right ) (-x-1)^{2/3} (1-x)^{2/3}}d\sqrt [3]{x}+\frac {4}{9} \int \frac {1}{\left (1-\sqrt [3]{3} \sqrt [3]{x}\right ) (-x-1)^{2/3} (1-x)^{2/3}}d\sqrt [3]{x}+\frac {4}{9} \int \frac {1}{\left (1-(-1)^{2/3} \sqrt [3]{3} \sqrt [3]{x}\right ) (-x-1)^{2/3} (1-x)^{2/3}}d\sqrt [3]{x}-\frac {\sqrt [3]{x} (x+1)^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{3},\frac {2}{3},\frac {4}{3},x,-x\right )}{(-x-1)^{2/3}}+\frac {\sqrt [3]{x} \left (1-x^2\right ) \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2-1}}\right ) \sqrt {\frac {\frac {x^{4/3}}{\left (x^2-1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{x^2-1}}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1-\frac {\left (1-\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{x^2-1}}}{1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{x^2-1}}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3 \sqrt [4]{3} (-x-1)^{2/3} (1-x)^{2/3} \sqrt {-\frac {x^{2/3} \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2-1}}\right )}{\sqrt [3]{x^2-1} \left (1-\frac {\left (1+\sqrt {3}\right ) x^{2/3}}{\sqrt [3]{x^2-1}}\right )^2}}}\right )}{\left (x^3-x\right )^{2/3}}\) |
Input:
Int[((-1 + x)*(1 + 3*x))/((-1 + 3*x)*(-x + x^3)^(2/3)),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.67 (sec) , antiderivative size = 606, normalized size of antiderivative = 4.85
| method | result | size |
| trager | \(\ln \left (-\frac {1415 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x -3679 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +1819 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-2127 \left (x^{3}-x \right )^{\frac {2}{3}}-3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}-2127 \left (x^{3}-x \right )^{\frac {1}{3}} x +1698 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+2572 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -712 x^{2}+2127 \left (x^{3}-x \right )^{\frac {1}{3}}-251 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-445 x -89}{-1+3 x}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {-1415 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x +3679 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +4649 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-1107 \left (x^{3}-x \right )^{\frac {2}{3}}-3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}-1107 \left (x^{3}-x \right )^{\frac {1}{3}} x -1698 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-4786 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -2522 x^{2}+1107 \left (x^{3}-x \right )^{\frac {1}{3}}+3145 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1552 x -1358}{-1+3 x}\right )+\ln \left (\frac {-1415 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x +3679 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +4649 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-1107 \left (x^{3}-x \right )^{\frac {2}{3}}-3234 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}-1107 \left (x^{3}-x \right )^{\frac {1}{3}} x -1698 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-4786 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -2522 x^{2}+1107 \left (x^{3}-x \right )^{\frac {1}{3}}+3145 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1552 x -1358}{-1+3 x}\right )\) | \(606\) |
Input:
int((-1+x)*(1+3*x)/(-1+3*x)/(x^3-x)^(2/3),x,method=_RETURNVERBOSE)
Output:
ln(-(1415*RootOf(_Z^2-_Z+1)^2*x^2+3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(2/3)+323 4*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3)*x-3679*RootOf(_Z^2-_Z+1)^2*x+1819*RootOf (_Z^2-_Z+1)*x^2-2127*(x^3-x)^(2/3)-3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3)-21 27*(x^3-x)^(1/3)*x+1698*RootOf(_Z^2-_Z+1)^2+2572*RootOf(_Z^2-_Z+1)*x-712*x ^2+2127*(x^3-x)^(1/3)-251*RootOf(_Z^2-_Z+1)-445*x-89)/(-1+3*x))*RootOf(_Z^ 2-_Z+1)-RootOf(_Z^2-_Z+1)*ln((-1415*RootOf(_Z^2-_Z+1)^2*x^2+3234*RootOf(_Z ^2-_Z+1)*(x^3-x)^(2/3)+3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3)*x+3679*RootOf( _Z^2-_Z+1)^2*x+4649*RootOf(_Z^2-_Z+1)*x^2-1107*(x^3-x)^(2/3)-3234*RootOf(_ Z^2-_Z+1)*(x^3-x)^(1/3)-1107*(x^3-x)^(1/3)*x-1698*RootOf(_Z^2-_Z+1)^2-4786 *RootOf(_Z^2-_Z+1)*x-2522*x^2+1107*(x^3-x)^(1/3)+3145*RootOf(_Z^2-_Z+1)+15 52*x-1358)/(-1+3*x))+ln((-1415*RootOf(_Z^2-_Z+1)^2*x^2+3234*RootOf(_Z^2-_Z +1)*(x^3-x)^(2/3)+3234*RootOf(_Z^2-_Z+1)*(x^3-x)^(1/3)*x+3679*RootOf(_Z^2- _Z+1)^2*x+4649*RootOf(_Z^2-_Z+1)*x^2-1107*(x^3-x)^(2/3)-3234*RootOf(_Z^2-_ Z+1)*(x^3-x)^(1/3)-1107*(x^3-x)^(1/3)*x-1698*RootOf(_Z^2-_Z+1)^2-4786*Root Of(_Z^2-_Z+1)*x-2522*x^2+1107*(x^3-x)^(1/3)+3145*RootOf(_Z^2-_Z+1)+1552*x- 1358)/(-1+3*x))
Time = 0.43 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.86 \[ \int \frac {(-1+x) (1+3 x)}{(-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=-\sqrt {3} \arctan \left (\frac {612314840 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (1609127381 \, x^{2} + 1235276981 \, x + 124616800\right )} + 2605939922 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{2990437623 \, x^{2} + 3108349623 \, x - 39304000}\right ) - \frac {1}{2} \, \log \left (\frac {3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 3 \, x - 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} - 1}{3 \, x - 1}\right ) \] Input:
integrate((-1+x)*(1+3*x)/(-1+3*x)/(x^3-x)^(2/3),x, algorithm="fricas")
Output:
-sqrt(3)*arctan((612314840*sqrt(3)*(x^3 - x)^(1/3)*(x - 1) + sqrt(3)*(1609 127381*x^2 + 1235276981*x + 124616800) + 2605939922*sqrt(3)*(x^3 - x)^(2/3 ))/(2990437623*x^2 + 3108349623*x - 39304000)) - 1/2*log((3*(x^3 - x)^(1/3 )*(x - 1) + 3*x - 3*(x^3 - x)^(2/3) - 1)/(3*x - 1))
\[ \int \frac {(-1+x) (1+3 x)}{(-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=\int \frac {\left (x - 1\right ) \left (3 x + 1\right )}{\left (x \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {2}{3}} \cdot \left (3 x - 1\right )}\, dx \] Input:
integrate((-1+x)*(1+3*x)/(-1+3*x)/(x**3-x)**(2/3),x)
Output:
Integral((x - 1)*(3*x + 1)/((x*(x - 1)*(x + 1))**(2/3)*(3*x - 1)), x)
\[ \int \frac {(-1+x) (1+3 x)}{(-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=\int { \frac {{\left (3 \, x + 1\right )} {\left (x - 1\right )}}{{\left (x^{3} - x\right )}^{\frac {2}{3}} {\left (3 \, x - 1\right )}} \,d x } \] Input:
integrate((-1+x)*(1+3*x)/(-1+3*x)/(x^3-x)^(2/3),x, algorithm="maxima")
Output:
integrate((3*x + 1)*(x - 1)/((x^3 - x)^(2/3)*(3*x - 1)), x)
\[ \int \frac {(-1+x) (1+3 x)}{(-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=\int { \frac {{\left (3 \, x + 1\right )} {\left (x - 1\right )}}{{\left (x^{3} - x\right )}^{\frac {2}{3}} {\left (3 \, x - 1\right )}} \,d x } \] Input:
integrate((-1+x)*(1+3*x)/(-1+3*x)/(x^3-x)^(2/3),x, algorithm="giac")
Output:
integrate((3*x + 1)*(x - 1)/((x^3 - x)^(2/3)*(3*x - 1)), x)
Timed out. \[ \int \frac {(-1+x) (1+3 x)}{(-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=\int \frac {\left (3\,x+1\right )\,\left (x-1\right )}{{\left (x^3-x\right )}^{2/3}\,\left (3\,x-1\right )} \,d x \] Input:
int(((3*x + 1)*(x - 1))/((x^3 - x)^(2/3)*(3*x - 1)),x)
Output:
int(((3*x + 1)*(x - 1))/((x^3 - x)^(2/3)*(3*x - 1)), x)
\[ \int \frac {(-1+x) (1+3 x)}{(-1+3 x) \left (-x+x^3\right )^{2/3}} \, dx=3 \left (\int \frac {x^{2}}{3 x^{\frac {5}{3}} \left (x^{2}-1\right )^{\frac {2}{3}}-x^{\frac {2}{3}} \left (x^{2}-1\right )^{\frac {2}{3}}}d x \right )-2 \left (\int \frac {x}{3 x^{\frac {5}{3}} \left (x^{2}-1\right )^{\frac {2}{3}}-x^{\frac {2}{3}} \left (x^{2}-1\right )^{\frac {2}{3}}}d x \right )-\left (\int \frac {1}{3 x^{\frac {5}{3}} \left (x^{2}-1\right )^{\frac {2}{3}}-x^{\frac {2}{3}} \left (x^{2}-1\right )^{\frac {2}{3}}}d x \right ) \] Input:
int((-1+x)*(1+3*x)/(-1+3*x)/(x^3-x)^(2/3),x)
Output:
3*int(x**2/(3*x**(2/3)*(x**2 - 1)**(2/3)*x - x**(2/3)*(x**2 - 1)**(2/3)),x ) - 2*int(x/(3*x**(2/3)*(x**2 - 1)**(2/3)*x - x**(2/3)*(x**2 - 1)**(2/3)), x) - int(1/(3*x**(2/3)*(x**2 - 1)**(2/3)*x - x**(2/3)*(x**2 - 1)**(2/3)),x )